Spatial Cognition and Computation 2: 181–218, 2000.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Representing space in a PDP network: Coarse
allocentric coding can mediate metric and nonmetric
MICHAEL R.W. DAWSON*, PATRICIA M. BOECHLER and
Biological Computation Project, Department of Psychology, University of Alberta,
Edmonton, Alberta, CANADA T6G 2P9 (*Author for correspondence: E-mail:
Received 22 June 2000; accepted 10 September 2001
Abstract. In one simulation, an artiﬁcial neural network was trained to rate the distances
between pairs of cities on the map of Alberta, given only place names as input. Distance
ratings ranged from 0 (when the network rated the distance between a city and itself) to 10.
The question of interest was the nature of the representations developed by the network’s six
hidden units after it successfully learned to make the desired responses. Analyses indicated
that the network used coarse allocentric coding to solve this problem. Each hidden unit could
be described as occupying a position on the map of Alberta, and each connection weight
feeding into a hidden unit was related to the distance on the map between the hidden unit
and one of the stimulus cities. On its own, a single hidden unit’s response was a relatively
inaccurate distance measure. However, by combining all six hidden unit responses in a coarse
coding scheme, accurate responses were generated by the network. In a second simulation,
a second network was trained to make similar judgements, but was trained to violate the
minimality constraint on metric space when trained to judge the distance between a city and
itself. An analysis of this network indicated that it too was using coarse allocentric coding.
Key words: artiﬁcial neural network, coarse allocentric coding, minimality principle,
symmetry principle, triangle inequality
Our everyday interactions with the visual and spatial world are grounded
in the essential experience that space is metric. Mathematically speaking,
a space is metric if relationships between locations or points in the space
conform to three different principles (Blumenthal 1953). The ﬁrst is the
minimality principle, which dictates that the shortest distance in the space
is between a point x and itself. The second is the symmetry principle,which
dictates that the distance in the space between two points x and y is equal to
the distance between points y and x. The third is the triangle inequality,which